More importantly, there exists a modification of the VAW forecaster due to [1]} that can remove the logarithmic factor in (REF ).We are thankful to Manfred Warmuth for pointing us to the modified VAW forecaster. Let us introduce the modified VAW forecaster.
Given a sample \(S_{n}\)  and any \(X \in {\mathbb {R}}^{d}\) , let \(h_{X} = X^{\mathsf {T}} (\sum _{i=1}^{n} X_{i}X_{i}^{\mathsf {T}} + XX^{\mathsf {T}})^{\dagger }X\)  denote the leverage of the point \(X\)  with respect to
the covariates \(X_{1}, \dots , X_{n}, X\) , where the notation \(A^{\dagger }\)  denotes the Moore-Penrose inverse of a matrix \(A\) . The modified VAW predictor \(\widehat{f}\)  is then defined pointwise as follows:
\(\widehat{f}(X) = (1 - h_{X})\widehat{f}^{\operatorname{VAW}}_{0}(X).\) 
